Editing Expectation–maximization algorithm (section)

=== α-EM algorithm ===
The Q-function used in the EM algorithm is based on the log likelihood. Therefore, it is regarded as the log-EM algorithm. The use of the log likelihood can be generalized to that of the α-log likelihood ratio. Then, the α-log likelihood ratio of the observed data can be exactly expressed as equality by using the Q-function of the α-log likelihood ratio and the α-divergence. Obtaining this Q-function is a generalized E step. Its maximization is a generalized M step. This pair is called the α-EM algorithm<ref>
{{cite journal
|last=Matsuyama |first=Yasuo
|title=The α-EM algorithm: Surrogate likelihood maximization using α-logarithmic information measures
|journal=IEEE Transactions on Information Theory
|volume=49 | year=2003 |pages=692–706 |issue=3
|doi=10.1109/TIT.2002.808105
}}
</ref>
which contains the log-EM algorithm as its subclass. Thus, the α-EM algorithm by [[Yasuo Matsuyama]] is an exact generalization of the log-EM algorithm. No computation of gradient or Hessian matrix is needed. The α-EM shows faster convergence than the log-EM algorithm by choosing an appropriate α. The α-EM algorithm leads to a faster version of the Hidden Markov model estimation algorithm α-HMM.
<ref>
{{cite journal
|last=Matsuyama |first=Yasuo
|title=Hidden Markov model estimation based on alpha-EM algorithm: Discrete and continuous alpha-HMMs
|journal=International Joint Conference on Neural Networks
| year=2011 |pages=808–816
}}
</ref>